In the paper, new Fritz John type necessary optimality conditions and new Karush--Kuhn--Tucker type necessary opimality conditions are established for the considered nondifferentiable multiobjective programming problem involving locally Lipschitz functions. Proofs of them avoid the alternative theorem usually applied in such a case. The sufficiency of the introduced Karush--Kuhn--Tucker type necessary optimality conditions are proved under assumptions that the functions constituting the considered nondifferentiable multiobjective programming problem are $G$-$V$-invex with respect to the same function $\eta$. Further, the so-called nondifferentiable vector $G$-Mond--Weir dual problem is defined for the considered nonsmooth multiobjective programming problem. Under nondifferentiable $G$-$V$-invexity hypotheses, several duality results are established between the primal vector optimization problem and its $G$-dual problem in the sense of Mond--Weir.